Mechanical Engineering Department. Auburn University ... surface contact, friction and wear. ..... Williams, J. A. Engineering Tribology, New York, Oxford, 2000.
Contact Modeling of Rough Surfaces Robert L. Jackson Mechanical Engineering Department Auburn University
Background • The
modeling of surface asperities on the micro-scale is of great interest to those interested in the mechanics of surface contact, friction and wear. • When considering the area of contact between real objects, the roughness of their surfaces must be accounted for, in that it will determine the real area of contact between them.
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Rough Surface Contact Models • Statistical – Model surface as a statistical distribution of asperities with various heights and properties (Computationally inexpensive). • Deterministic – Model the real features of the surface as with much detail as possible (Computationally expensive). • FFT Methods: Problem solved in Frequency domain. • Fractal: Multiple scale roughness is considered.
Statistical Contact Model (Greenwood & Williamson)
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Hertz Contact Solution (1882) • Closed-form expressions to the deformations and stresses of two spheres in a purely elastic contact (Theory of Elasticity). • The Hertz solution assumes that the interference is small enough such that the geometry does not change significantly. • The solution also approximates the sphere surface as a parabolic curve with an equivalent radius of curvature at its tip. • It is also assumed that the contact surfaces are frictionless.
Hertz Solution Results
1 1 − ν 12 1 − ν 22 = + E′ E1 E2
AE = πRω
1 1 1 = + R R1 R2
PE =
4 E ′ R (ω ) 3 / 2 3
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Fully Plastic Truncation Model
AP = 2πRω
Pp = 2πRωH
H = 3⋅Sy
Hardness • The average contact pressure (P/A) when a contact surface has fully yielded (the entire contact surface is plastically deforming). • Usually assumed to be approximately 3⋅Sy as predicted by slip-line theory (Tabor, 1951). • However, Williams (1994) suggests a hardness value of 2.83⋅ Sy. • Hardness is not an independent material property and is dependant on the (deformed) contact geometry, as well as E, Sy, ν.
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Critical Interference (Initial Yielding) • Using the von Mises Yield Criteria and the Hertz Contact solution the following numerically fit solution is obtained.
⎛π ⋅C ⋅ Sy ω c = ⎜⎜ ⎝ 2E ′
2
⎞ ⎟⎟ R ⎠
C = 1.295 exp(0.736ν )
Normalization ω * = ω / ωc
AE* = ω *
( )
P = ω * E
* 3/ 2
P * = P / Pc
A* = A / Ac
* AAF = 2ω * * PAF =
3H * ω CS y
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CEB Model • CEB model (Chang et al., 1987) approximates elastoplastic contact by modeling a plastically deformed portion of a hemisphere using volume conservation. • Assumes average contact pressure to be constant hardness once yielding occurs. • Discontinuity at critical interference. • For Elasto-Plastic Deformation:
(
ACEB = πRω 2 − 1 / ω *
(
)
)
PCEB = πRω 2 − 1 / ω * KH
ZMC Model • ZMC model (Zhao et. al. 2000) interpolates between the elastic and fully plastic models. • A template function satisfies continuity of the function and its slope at the two transitions.
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FEM Elasto-plastic Model • Kogut and Etsion (2002) performed a FEM analysis of the same case of an elastic-perfectly plastic sphere in contact with a rigid flat. • In this analysis, the value of H is set to be fixed at 2.8⋅Sy. • Very similar to current model, although the finite element mesh used is much more course than the current mesh.
Spherical Contact Model
Just Before Contact
Mostly Elastic Deformation
Mostly Plastic Deformation
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Finite Element Model • Perfectly plastic material yields according to the von Mises yield criterion. • 100 Contact Elements are used to model the contact at the interface between the sphere and the rigid flat. • Iterative scheme used to relax problem to convergence. • Mesh convergence was satisfied.
